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24 II REAL NUMBERS

I c I From (IV) it follows that the intervals I_n have a nonempty inter-
sotion J- if J contained at least two distinct elements a < /?, the interval
h ft would be contained in each !„, and therefore by (2.2.14) we should
fia've 2-^B$*j?-a, or 1^2ⁿ(p-a) for every n, which contradicts (III)
(remember that T^n, as is obvious by induction). Therefore J = {y}.
Let us first show that y is a majorant of X; if not, there would be an x e X
such that x>y; but there would then be an n such that 2~~ⁿ < x - y, and
as ye I,, we would have a+p_n2~ⁿ<x₉ contrary to the definition of /?„.
On the other hand, every yeM is ^y; otherwise, there would be an
n such that 2~ⁿ < y - y, and as y e l_n, we would have a + (/>„- 1)2"" > y,
and a + (p_n - 1)2~" would be a majorant of X; this contradicts again the
definition of p_n. The number y is thus the smallest element of M; it is called
the least upper bound or supremum of X, and written l.u.b. X, or sup X.

(2.3.3) If a nonempty subset XofR is minorized, the set of minor ants M'
ofX has a greatest element.

Apply (2.3.2) to the set -X.

The greatest element of M' is called the greatest lower bound or infimum
of X and written g.l.b. X or inf X. For a nonempty bounded set X, both
inf X and sup X exist, and inf X < sup X.

(2,3.4.) The l.u.b. of a majorized set X is the real number y characterized
by the following two properties: (1) y is a majorant ofX; (2) for every integer
n> 0, there exists an element xeX such that y — l/n < x < y.

Both properties of y = sup X follow from the definition, since the second
expresses that y - l/n is not a majorant of X. Conversely, if these properties
are satisfied, we cannot have sup X = /? < y, for there would be an n such
that 1//2 < y- ft, hence fi<y- l/n, and y - l/n would be a majorant of X»
contrary to property (2). A similar characterization holds for inf X, by
applying (2.3.4) to -X, since inf X = -sup(-X).

If a set X c E. has & greatest element b (resp. a smallest element a), then
b = sup X (resp. a = inf X) and we write max X (resp. min X) instead of
sup X (resp. inf X). This applies in particular to finite sets by (2.2.3), But
the l.u.b. and g.l.b. of a bounded infinite set X need not belong to X; for
instance, if X is the set of all numbers l/n, where n runs through all integers
>l,Oistheg.l.b.ofX.

(2.3.5) If A c R is majorized and B c A, B is majorized and sup B < sup A.
This follows from the definitions.



	24 II REAL NUMBERS I c I From (IV) it follows that the intervals I_n have a nonempty inter- sotion J- if J contained at least two distinct elements a < /?, the interval h ft would be contained in each !„, and therefore by (2.2.14) we should fia've 2-^B$j?-a, or 1^2ⁿ(p-a)* for every n, which contradicts (III) (remember that T^n, as is obvious by induction). Therefore J = {y}. Let us first show that y is a majorant of X; if not, there would be an x e X such that x>y; but there would then be an n such that 2~~ⁿ < x - y, and as ye I,, we would have a+p_n2~ⁿ<x₉ contrary to the definition of /?„. On the other hand, every yeM is ^y; otherwise, there would be an n such that 2~ⁿ < y - y, and as y e l_n, we would have a + (/>„- 1)2"" > y, and a + (p_n - 1)2~" would be a majorant of X; this contradicts again the definition of p_n. The number y is thus the smallest element of M; it is called the least upper bound or supremum of X, and written l.u.b. X, or sup X. (2.3.3) If a nonempty subset XofR is minorized, the set of minor ants M' ofX has a greatest element. Apply (2.3.2) to the set -X. The greatest element of M' is called the greatest lower bound or infimum of X and written g.l.b. X or inf X. For a nonempty bounded set X, both inf X and sup X exist, and inf X < sup X. (2,3.4.) The l.u.b. of a majorized set X is the real number y characterized by the following two properties: (1) y is a majorant ofX; (2) for every integer n> 0, there exists an element xeX such that y — l/n < x < y. Both properties of y = sup X follow from the definition, since the second expresses that y - l/n is not a majorant of X. Conversely, if these properties are satisfied, we cannot have sup X = /? < y, for there would be an n such that 1//2 < y- ft, hence fi<y- l/n, and y - l/n would be a majorant of X» contrary to property (2). A similar characterization holds for inf X, by applying (2.3.4) to -X, since inf X = -sup(-X). If a set X c E. has & greatest element b (resp. a smallest element a), then b = sup X (resp. a = inf X) and we write max X (resp. min X) instead of sup X (resp. inf X). This applies in particular to finite sets by (2.2.3), But the l.u.b. and g.l.b. of a bounded infinite set X need not belong to X; for instance, if X is the set of all numbers l/n, where n runs through all integers >l,Oistheg.l.b.ofX. (2.3.5) If A c R is majorized and B c A, B is majorized and sup B < sup A. This follows from the definitions.